The Price of Inflation and Foreign Exchange Risk
in International Equity Markets
Cesare Robotti
Working Paper 2001-26
November 2001
Working Paper Series
Federal Reserve Bank of Atlanta
Working Paper 2001-26
November 2001
The Price of Inflation and Foreign Exchange Risk
in International Equity Markets
Cesare Robotti, Federal Reserve Bank of Atlanta
Abstract: In this paper the author formulates and tests an international intertemporal capital asset pricing
model in the presence of deviations from purchasing power parity (II-CAPM [PPP]). He finds evidence in favor
of at least mild segmentation of international equity markets in which only global market risk appears to be
priced. When using the Hansen & Jagannathan (1991, 1997) variance bounds and distance measures as testing
devices, the author finds that, while all international asset pricing models are formally rejected by the data,
their pricing implications are substantially different. The superior performance of the II-CAPM (PPP) is mainly
attributable to significant hedging against inflation risk.
JEL classification: G12, G15
Key words: international intertemporal capital asset pricing model, purchasing power parity, hedging demands
The author thanks Pierluigi Balduzzi, Arthur Lewbel, Shijun Liu, and seminar participants at Boston College, the University
of California at Irvine, and the 2000 Meetings of the Financial Management Association for useful comments. The views
expressed here are the author’s and not necessarily those of the Federal Reserve Bank of Atlanta or the Federal Reserve
System. Any remaining errors are the author’s responsibility.
Please address questions regarding content to Cesare Robotti, Financial Economist, Research Department, Federal Reserve
Bank of Atlanta, 1000 Peachtree Street, N.E., Atlanta, Georgia 30309-4470, 404-498-8543, 404-498-8810 (fax),
[email protected].
The full text of Federal Reserve Bank of Atlanta working papers, including revised versions, is available on the Atlanta Fed’s
Web site at http://www.frbatlanta.org/publica/work_papers/index.html. To receive notification about new papers, please
use the on-line publications order form, or contact the Public Affairs Department, Federal Reserve Bank of Atlanta, 1000
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Introduction
In recent years, several papers have tested the international version of the Capital Asset
Pricing Model (I—CAPM). Some of these papers have found that in addition to compensation
for global market risk, investors require compensation for inflation and foreign exchange risk.
However, the evidence on the ability of different specifications of the I—CAPM to correctly
price international asset returns is mixed. Dumas and Solnik (1995) and De Santis and
Gérard (1998) cannot reject a specification of the I—CAPM that includes global market risk
and foreign exchange risk. Their analysis is consistent with the hypothesis of integration of
international equity and foreign exchange markets. On the contrary, Vassalou (2000) rejects
the adequacy of several nested versions of the I—CAPM with foreign exchange and inflation
risks to explain most of the cross—sectional variation in stock excess returns. Her work seems
to support the hypothesis of at least mild segmentation in international stock markets.
Moreover, the evidence regarding the size and significance of the economic risk premia is
less than conclusive. While there is some evidence of time variation in the premia, the patterns of time variation are somewhat unclear. For example, De Santis and Gérard (1997,1998)
find that exposure to global market risk commands a positive and significant average risk
premium only when a conditional, fully parametric specification of the I—CAPM with time—
varying price of risk is assumed. Similarly, Ferson and Harvey (1993) find a positive and
significant risk premium for exposure to global market risk in a time—varying multi—beta international asset pricing model. Dumas and Solnik (1995) provide evidence of time variation
in the global market risk premium, but do not report size and statistical significance of the
average conditional market premium. Vassalou (2000) does not comment on the sign and the
significance of the unconditional global market premium. The evidence on the premia associated with foreign exchange risk and inflation risk is also unclear. For example, De Santis
and Gérard (1998) find support for a specification of the conditional I—CAPM that includes
both global market risk and foreign exchange risk under the assumption of time variation in
all prices of risk. Their results are globally consistent with the findings of Dumas and Solnik
(1995). Even if both studies assume time—varying prices of risk, the main difference between
1
the two approaches is that Dumas and Solnik (1995) test an unconditional version of the
conditional I—CAPM using the generalized method of moments, while De Santis and Gérard
(1998) explicitly model time variation in the first two conditional moments using VGARCH
specifications. On the other hand, Vassalou (2000) finds strong support for versions of the
I—CAPM that take inflation risk into account, mixed evidence for versions of the I—CAPM
with foreign exchange risk, and no support for international asset pricing models where
both foreign exchange premia and inflation premia are jointly estimated. In addition, the
magnitudes of the estimated risk premia change substantially from one study to the other.
Two main issues emerge from the previous discussion. First, it is hard to reconcile
home bias in international equity and foreign exchange markets with the findings that stock
markets are perfectly integrated and that the I—CAPM holds. One of the reasons why the
I—CAPM holds in some studies and not in others might be that the testing procedure used
crucially depends on the level of aggregation of the equity portfolios used in the analysis. For
example, Vassalou (2000) considers not only cross—country variation but also within—country
variation in equity returns in order to control for home bias. On the other hand, Dumas
and Solnik (1995) and De Santis and Gérard (1998) use aggregated stock return data from
Morgan Stanley Capital International (MSCI). In the effort to control for home bias, the
authors include in the analysis idiosyncratic risks and country—specific effects. Nonetheless,
De Santis and Gérard (1997,1998) reach a quite puzzling result. They find that when the
restriction of non—negativity of the conditional global market risk premium is imposed, there
is some residual predictability in stock excess returns that can be explained by idiosyncratic
risk and country—specific effects. On the other hand, when the restriction of non—negativity
is relaxed, this residual predictability disappears and they cannot reject the hypothesis of
integration of international stock markets. Second, the patterns of time variation in the first
two conditional moments of excess asset returns and in the prices of risk are not clear. The
I—CAPM in its unconditional version does not seem to hold.1 It also does not seem to hold
in its conditional version unless all prices of risk are assumed to be time—varying.2 I try to
1
2
See, for example, Solnik (1974), and Stehle (1977).
See, for example, Hodrick (1981), De Santis and Gérard (1998), Dumas and Solnik (1995), and Bekaert
and Harvey (1995).
2
clarify the above-mentioned two issues by separating the problem of economic risk premia
estimation from the problem of testing intertemporal international asset pricing models and
by implementing a convenient new approach to the estimation of economic risk premia. I
also implement a comprehensive set of tests of asset pricing models with foreign exchange
and inflation risks.
This paper contributes to the existing literature in three different ways. First, I use a new
methodology to estimate the inflation and foreign exchange risk premia which is based on
the minimum-variance stochastic discount factors of Hansen and Jagannathan (1991) (HJ).
This pricing kernel prices, by construction, the asset returns under consideration, and has
the minimum variance among all kernels consistent with asset returns. The economic risk
premia that I estimate are those assigned by the minimum—variance kernel. This approach
has several advantages on the traditional methods:
• I do not need to specify all relevant sources of risk; i.e., the estimate of a risk premium
does not change depending on which other sources of uncertainty are simultaneously
considered.
• The estimates are robust to the form of the pricing kernel. Any admissible pricing
kernel has the same price implications and, hence, assigns the same risk premia as the
HJ minimum—variance kernel.
• The estimation procedure is quite simple. I estimate the parameters of the minimum—
variance kernel and the covariance of the kernel with the different sources of uncertainty.
By comparison, studies based on multi—beta models require the estimation of linear
factor models for each of the securities, and then the estimation of the risk premia.
• I show that when the non—negativity restriction on the pricing kernel is not imposed,
the proposed economic risk premia reduce to the expected cash flows on the portfolios
that best hedge the risk factors, financed at the riskless rate. This result makes it clear
that the risk premia estimates crucially depend on the hedging portfolios composition,
and hence on the assets available for investment.
3
Second, I model the time variation of economic risk premia by specifying a pricing kernel
linear in the set of instruments used in the analysis. Finally, I use a new methodology to
test the international CAPM, which is based on the construction of portfolios mimicking the
behavior of global market risk, foreign exchange, and inflation risks. Specifically, I test the
International Static CAPM (IS-CAPM), the International CAPM with demands hedging
against foreign exchange and inflation risks (I-CAPM), and the International Intertemporal
CAPM (II-CAPM) by using a variety of diagnostics: the standard J—test of overidentifying
restrictions of Hansen and Singleton (1982); the Hansen—Jagannathan (1991) variance bound;
and the Hansen—Jagannathan (1997) distance measure. I am not aware of any formal test
of the I—CAPM with demands hedging against variations in the investment opportunity set
of an international investor in presence of deviations from Purchasing Power Parity (PPP)
and currency risk.3
The main results of the paper can be summarized as follows: First, I show that only
global market risk commands a highly positive and significant unconditional risk premium
as indicated by its relative Sharpe ratio. Inflation risk does not seem to be priced both unconditionally and conditionally, while global market and exchange risks exhibit interesting
patterns of time variation. Second, none of the international asset pricing models are rejected
by the data when, following Hansen and Singleton (1982), I perform tests of the overidentifying restrictions that they impose. On the contrary, testing the same specifications with
the HJ variance bounds and distance measures leads to a strong rejection of all models.4
Hence, even if all international asset pricing models deliver different pricing implications,
none of them seems to hold when the testing procedure adopted is stringent enough. Large
and statistically significant inflation-hedging demands explain the better pricing implications
delivered by the II-CAPM in presence of deviations from PPP. Interestingly, this analysis
is able to rationalize recent findings in the international asset pricing literature and, at the
same time, shed some light on the controversial home bias puzzle. Even if I cannot explain
3
Hodrick, Ng, and Sengmüller (1999) test an international version of Campbell’s (1996) intertemporal
CAPM, but do not consider deviations from PPP.
4
See Zhang (2001) for an evaluation of domestic and international asset pricing models using the HJ
(1997) distance measure.
4
home bias in international equity markets, I can argue that my results are consistent with
the hypothesis of at least mild segmentation of international stock markets.
The remainder of this paper is organized as follows: Section I discusses the economic
risk premia assigned by the minimum—variance admissible kernel and explicit international
asset pricing models;5 Section II illustrates the methods used for estimation and testing;
Section III describes the data; Section IV identifies the investment opportunity set of an
investor; Section V presents the results of the risk premia estimation and relative Sharpe
ratios; Section VI presents the results of the tests of international asset pricing models and
empirically identifies the size of the hedging demands; and Section VII concludes.
I.
Methodology: Economic Risk Premia and International Asset Pricing Models
This section derives a general pricing result for nominal asset returns denominated in terms
of a reference currency. Namely, I show that the expected nominal returns in excess of
the nominal risk-free rate of the reference country are directly related to the covariance
of a nominal pricing kernel, scaled by the risk-free rate, with nominal asset returns. I then
consider economic variables that are believed to be relevant for international asset prices: the
rate of return on the world market portfolio; the rate of appreciation of the reference currency
relative to the other currencies; and the rates of inflation in the different countries. If I can
construct portfolios that exactly replicate the conditional variability of these variables, then
the cross moments of the scaled (or normalized) pricing kernel translate into risk premia.
Specifically, these cross moments reduce to the expected nominal cash flows on zero-netinvestment positions long in the mimicking portfolio and short in the riskless asset. If the
nominal (normalized) pricing kernel is linear in the economic variables described above,
as is often assumed in international asset pricing models, then the risk premia also enter
familiar “beta” pricing results. In addition, I show how to test several specifications of the
5
The analysis of risk premia mainly follows Balduzzi & Kallal (1997) and Balduzzi & Robotti (2001).
5
international capital asset pricing model using a variety of diagnostics.
A.
A General International Asset Pricing Result
Assume there are L + 1 countries and a set of N = n + 1 assets, other than the measurementcurrency nominally risk-free asset. These include n risky assets, or portfolios of risky assets,
and the world portfolio of risky assets.
Consider now the N × 1 vector of (gross) nominal returns on the N assets, r. From now
on, I shall assume all returns to be denominated in terms of the reference currency. By the
law of one price, I have
Et (mt+1 rt+1 ) = 1
(1)
for some admissible nominal pricing kernel m,6 where 1 is an N × 1 vector of ones. Note that
the restriction stated in equation (1) prevents arbitrage in the security markets, but not in
the commodity markets where Purchasing Power Parity (PPP) may not hold.
Let rf t denote the nominally risk-free rate of the reference country. I then have
Et (mt+1 rf t ) = 1 .
(2)
It is convenient to perform the analysis that follows in terms of the pricing kernel scaled by the
risk-free rate. That is, I use the normalized pricing kernel, mt+1 rf t ≡ qt+1 , where Et (qt+1 ) =
1. As a consequence, I circumvent the problem of explicitly modeling the conditional mean
of the pricing kernel m.
Using equations (1) and (2), I obtain the familiar orthogonality conditions
Et [qt+1 (rt+1 − 1rft )] = 0 .
(3)
Et (qt+1 )[Et (rt+1 ) − 1rft ] = Et (rt+1 ) − 1rft = −Covt (qt+1 , rt+1 ) .
(4)
Rearranging, I obtain
6
See Harrison and Kreps (1979). The set of pricing kernels can be interpreted as the set of nominal
intertemporal marginal rates of substitution compatible with the distribution of returns.
6
The conditional risk premium on any asset equals the opposite of the conditional covariance
between the normalized pricing kernel and the asset return.
B.
Economic Variables and Exact Mimicking Portfolios
Consider now a K × 1 vector of economic variables y. Without loss of generality, I assume
these variables to have constant zero mean and unit variance: i.e., Et (yk,t+1 ) = 0 and
2
) = 1, for k = 1, . . . , K.
Et (yk,t+1
The reason for assuming a constant zero mean is that security markets only price unanticipated variability; the relevant economic variables are, in fact, innovations. Finally, the
reason for scaling the risk factors to have constant unit variance is that I can compare the
“prices” attached to the economic variables without having to worry about differences in
variability.
Assume now that there exist portfolios that exactly mimic the behavior of the risk factors.
Let ry,t+1 ≡ yt+1 , denote the K × 1 vector of mimicking portfolio returns. According to (4),
I have
Et (ry,t+1 ) − rf t 1 = −Et [yt+1 (qt+1 − 1)] ≡ λyt ,
(5)
which is the vector of conditional risk premia on the corresponding economic variables.
Hence, the conditional cross moments between the normalized pricing kernel q and the
economic variable yk equals, with the opposite sign, the conditional risk premium on the
variable. Assuming stationarity and using the law of iterated expectations, I have
λy ≡ E(λyt ) ≡ −E[(qt+1 − 1)yt+1 ] ,
which is the vector of unconditional or mean risk premia on y. The unconditional cross
moment between the normalized pricing kernel q and the economic variable yk equals, with
the opposite sign, the mean risk premium on the variable.
7
C.
The Minimum-Variance Kernel
Using the definition of the normalized pricing kernel q, I can rewrite equations (1) and (2)
as
Et (qt+1 rt+1 ) = rf t 1
(6)
Et (qt+1 ) = 1 .
(7)
and
Following HJ, I can construct an admissible (normalized) pricing kernel, i.e., a random
?
variable that satisfies equations (6) and (7), that is linear in rt+1 : qt+1
≡ α0t + r>
t+1 αt , where
α0t is a scalar and αt is an N × 1 coefficient vector. I then have
Et (qt+1 )Et (rt+1 ) + Covt (qt+1 , rt+1 ) = rf t 1 .
(8)
?
Using (6) and qt+1
≡ α0t + r>
t+1 αt , I have
−Σrrt αt = Et (rt+1 ) − rf t 1 ,
(9)
where Σrrt is the conditional covariance matrix of risky asset returns (which I assume to be
invertible). I obtain
αt = −Σ−1
rrt [Et (rt+1 ) − rf t 1]
(10)
α0t = 1 − Et (rt+1 )> αt .
(11)
while
Using the results in equations (10) and (11), I have
?
qt+1
= 1 − [rt+1 − Et (rt+1 )]> Σ−1
rrt [Et (rt+1 ) − rf t 1] .
(12)
?
This minimum-variance normalized pricing kernel, qt+1
, has several properties worth
noting. First, the vector αt is proportional to the vector of portfolio weights of the tangency
8
?
portfolio obtained from the risky security returns rt+1 .7 Hence, qt+1
is perfectly negatively
correlated with the rate of return on the tangency portfolio, rτ,t+1 . Another important
?
property is that the conditional variance of qt+1
equals the squared conditional Sharpe ratio
of the tangency portfolio, Sτ t . I have
?
?
[Covt (qt+1
, rτ,t+1 )]2 = Vart (rτ,t+1 )Vart (qt+1
).
(13)
?
Since qt+1
correctly prices all the securities under consideration, it also correctly prices the
?
tangency portfolio, and I have −Covt (qt+1
, rτ,t+1 ) = Et (rτ,t+1 ) − rf t . Using this result and
rearranging equation (13) above, I obtain

2
E (r
) − rf t 
 t τ,t+1
q
Vart (rτ,t+1 )
?
≡ S2τ t = Vart (qt+1
).
(14)
?
?
?
?
Finally, since Vart (qt+1
) = Et [(qt+1
− 1)2 ], then Var(qt+1
) = E[(qt+1
− 1)2 ], and
?
Var(qt+1
) = E(S2τ t ) .
(15)
?
Hence, the unconditional variance of qt+1
equals the average squared Sharpe ratio of the
tangency portfolio.
D.
Hedging Portfolios and Risk Premia
Now consider the conditional projection of the risk variable yk,t+1 on a constant and rt+1
?
(yk,t+1
≡ α0yk t + r>
t+1 αyk t , where α0yk t is a scalar and αyk t is an N × 1 coefficient vector).
?
?
?
?
) = −Covt (qt+1
yk,t+1
). Since qt+1
satisfies the conditional moment
Let λ?kt ≡ −Covt (qt+1 yk,t+1
restriction (6) by construction, I have
?
?
?
λ?kt = −Covt (qt+1
, yk,t+1
) = −Covt (qt+1
, α>
yk t rt+1 )
?
>
= −α>
yk t Covt (qt+1 , rt+1 ) = αyk t [Et (rt+1 ) − rf t 1] ,
(16)
which is the mean cash flow generated by the portfolio hedging the risk variable yk financed
at the riskless rate. In other words, λ?kt is the risk premium on the hedging portfolio for the
7
For example, see Ingersoll (1987), p. 89.
9
risk variable yk,t+1 . This implies that λ?kt does not depend on the choice of the (normalized)
pricing kernel, but only on the asset returns under scrutiny. If the conditional volatility of
yk,t+1 equals one and the factor is exactly replicated by asset returns, then λ?kt is also the
conditional Sharpe ratio on the exact hedging portfolio.
The hedging portfolios defined here are analogous to the ”economic tracking portfolios”
of Lamont (2001). The main difference in his approach is that the portfolios are constructed
to track changes in expectations of future realizations of the economic variables. Instead,
the proposed hedging portfolios are designed to track the contemporaneous realizations of
economic variables.
The composition of the hedging portfolios is worth further discussion. First, the hedging
portfolios contain allocations to the riskless asset in the amount α0yk t /rf t . (The rate of
return on the hedging portfolio contains a constant component, α0yk t , resulting from the
investment in the riskless asset.) Second, the hedging portfolio quantities do not sum up to
one. In order to have a “true” hedging portfolio I have to scale the coefficients α0yk t and
αyk t by α0yk t /rf t + 1> αyk t . Finally, the composition of the hedging portfolios corresponds
to the coefficients of a regression of the risk factors on the asset returns
>
>
−1
[α0yk t , α>
yk t ] = [Et (rt+1 rt+1 )] Et (rt+1 yk,t+1 ) .
(17)
Hence, the hedging portfolios are the discrete-time counterparts of the portfolios held by a
dynamic portfolio optimizer to hedge against changes in the investment-opportunity set.8
Two properties relating to the estimation of the risk premia are also worth noting. First,
while the conditional expected excess cash flow on a mimicking portfolio equals the conditional covariance between the factor and the minimum-variance kernel, the realized excess
?
cash flow on the mimicking portfolio, α>
yk t (rt+1 −rf t 1), in general differs from −(qt+1 −1)yk,t+1 .
In fact,
>
−1
> −1
Et (yk,t+1 r>
t+1 )Et (rt+1 rt+1 ) (rt+1 − rf 1) 6= [rt+1 − Et (rt+1 )] Σrrt [Et (rt+1 ) − rf t 1]yk,t+1 . (18)
This observation is important because the estimates of the risk premia may differ in small
8
See Merton (1973).
10
samples, depending on which approach is used, and the precision of the estimates may
also differ. Second, the conditional Sharpe ratios of the hedging portfolios, Syk? t , depend
on how closely a risk variable is replicated. Since the replication is not perfect, in general
?
) < Vart (yk,t+1 ). This means that
Vart (yk,t+1
S2yk? t =
(λ?kt )2
(λ?kt )2
>
.
?
Vart (yk,t+1
)
Vart (yk,t+1 )
This observation is important because a hedging portfolio might receive a “small” risk premium and yet command a high Sharpe ratio. This is because it only captures a small fraction
of the variability of a risk variable.
Appendix A relates my approach to the estimation of economic risk premia to two alternative approaches widely considered in the literature: 1) Principal components approach;
and 2) Multi—beta models approach. The first approach assumes that a number of unobservable factors drive the variation in asset returns. The realizations of these factors, while
not directly observable, can be inferred from the statistical properties of asset returns. The
second approach explicitly identifies the factors with observable macro-economic variables,
and assumes the pricing kernel and/or asset returns to be linear in the factors.
E.
Hedging Portfolios and Linear Kernels
In this section, I describe the link between hedging portfolios and linear kernels. This link is
then used in Section F to formulate and test competitive international asset pricing models.
>
) ≡ Σyyt = I. Consider
Without loss of generality, I assume Et (yt+1 ) = 0 and Et (yt+1 yt+1
now an admissible kernel qy,t+1 which is linear in the risk variables
>
qy,t+1 = 1 − yt+1
λt .
(19)
Et (rt+1 ) − 1rf t = β t λt ,
(20)
Equation (4) above becomes
>
where β t = Et (rt+1 yt+1
) ≡ Σryt . This implies that returns satisfy the linear factor model
rt+1 = 1rf t + β t λt + β t yt+1 + ²t+1 ,
11
(21)
where λt is a K×1 vector of conditional risk premia and ²t+1 is an N ×1 vector of disturbances
orthogonal to yt+1 .
Now consider the projection of the risk variables onto the span of asset returns augmented
?
≡ α0yt + αyt r>
of a unit-constant. I have yt+1
t+1 , where α0yt is a K × 1 vector and αyt is a
K × N coefficient matrix. I have
αyt = Σyrt Σ−1
rrt ,
(22)
α0yt = −αyt Et (rt+1 ) ,
(23)
?
yt+1
≡ αyt [rt+1 − Et (rt+1 )] .
(24)
and
?
If, in the admissible linear pricing kernel qt+1 , I replace yt+1 with yt+1
, the pricing result
(20) does not change. In other words, the projection of qy,t+1 onto the augmented span of
?
asset returns, qy? ,t+1 ≡ (yt+1
)> λt , is also an admissible pricing kernel.
Consider now the minimum-variance kernel constructed based on the cash flows of the
hedging portfolios qy?? ,t+1 . This is the kernel with minimum variance that correctly prices
the hedging portfolios. I have
?
qy?? ,t+1 = 1 − (yt+1
)> Σ−1
y? y ? t αyt [Et (rt+1 ) − rf t 1] ,
(25)
where Σy? y? t is the covariance matrix of the hedging-portfolio cash-flows. It is straightforward
to verify that
Σy? y? t = Σyrt Σ−1
rrt Σryt
(26)
holds.9
Hence,
9
?
−1
−1
qy?? ,t+1 = 1 − (yt+1
)> (Σyrt Σ−1
rrt Σryt ) Σyrt Σrrt [Et (rt+1 ) − rf t 1] .
(27)
I have
−1
−1
−1
Σy? y? t = Et [αyt (rt+1 − Et (rt+1 ))(rt+1 − Et (rt+1 ))> α>
yt ] = Σyrt Σrrt Σrrt Σrrt Σryt = Σyrt Σrrt Σryt .
12
Under the null of the multi-beta model, Et (rt+1 ) − rf t 1 = Σryt λt , and the expression above
simplifies to
?
qy?? ,t+1 = 1 − (yt+1
)> λt .
(28)
In other words, under the null, the projection of qy,t+1 onto the augmented span of asset
returns equals the minimum-variance kernel constructed using the hedging portfolio cashflows.
F.
Hedging Portfolios and Tests of Asset-Pricing Models
One appealing property of the kernel qy?? ,t+1 is that, under the null, it is the minimumvariance admissible kernel. In other words, any other admissible kernel is at least as volatile
as qy?? ,t+1 . This minimum-volatility property is obviously appealing. When it comes to tests
of overidentifying restrictions (the standard J test of Hansen, 1982), low volatility of the
kernel implies low volatility of the pricing errors, qy?? ,t+1 rt+1 − rf t 1. This, in turn, reduces
the likelihood that a poor model is not rejected simply because the volatility of the candidate
pricing kernel is high.
One additional feature that makes the kernel qy?? ,t+1 appealing is that the statistics associated with tests of the Hansen-Jagannathan variance bounds (HJV) and the HansenJagannathan distance (HJD) are the same, and can be interpreted in terms of comparisons
of expected squared conditional Sharpe ratios of the unrestricted tangency portfolio and a
restricted tangency portfolio.
The unconditional variance of qy?? ,t+1 equals the expectation of the squared conditional
Sharpe ratio of the tangency portfolio obtained using the hedging portfolios, instead of all
the assets available. Hence, when I test whether qy?? ,t+1 satisfies the HJ variance bound, I test
?
whether Var(qt+1
) − Var(qy?? ,t+1 ) > 0. This is equivalent to a test that E(S2τ t ) − E(S2y? τ t ) > 0
where Sy? τ t is the Sharpe ratio of the restricted tangency portfolio.
The HJD test is a test that the projection of a candidate kernel onto the augmented span
of asset returns and the minimum-variance kernel are sufficiently “close”: under the null, the
13
second moment of the difference between the minimum-variance kernel and the projection
of the candidate kernel onto the augmented span of asset returns should be “small”. In
particular, using qy? ,t+1 as the candidate kernel (note that the projection of qy?? ,t+1 onto a
constant and rt+1 is qy?? ,t+1 itself) the HJD statistic is the sample counterpart of
?
?
?
E[(qt+1
− qy?? ,t+1 )2 ] = E{Et (qt+1
− qy?? ,t+1 )2 } = E[Vart (qt+1
− qy?? ,t+1 )] .
(29)
Now note that
?
> −1
Covt (qt+1
− qy?? ,t+1 ) = [Et (rt+1 − rf t 1)]> Σ−1
rrt Σrr αy Σrr αy [Et (rt+1 ) − rf t 1]
−1
= [Et (rt+1 ) − rf t 1]> α>
y Σrr αy [Et (rt+1 ) − rf t 1]
= Vart (qy?? ,t+1 ) .
(30)
Hence
?
?
E[Vart (qt+1
− qy?? ,t+1 )] = E[Vart (qt+1
)] − E[Vart (qy?? ,t+1 )]
?
= Var(qt+1
) − Var(qy?? ,t+1 ) .
(31)
Up to this point, I have assumed the existence of a pricing kernel qt+1 which prices exactly the securities under consideration. I now investigate theories which postulate the form
of qt+1 . For concreteness, I consider here four models: the International Static CAPM (IS—
CAPM); the International CAPM in presence of deviations from PPP (I-CAPM (PPP)); the
International CAPM in presence of currency risk (I-CAPM (SPOT)); and the International
Intertemporal CAPM (II—CAPM). The IS—CAPM was first derived by Solnik (1974) and postulates a linear relationship between the cross-section of international expected excess equity
returns and the excess return on a world market portfolio. The I-CAPM, as introduced by
Adler and Dumas (1983), links nominal excess returns on international equity denominated
in a reference currency to a world market portfolio and portfolios hedging against deviations
from PPP. The II-CAPM is a combination of Merton’s (1973) intertemporal CAPM and
Adler and Dumas’s (1983) international CAPM. Namely, the cross-section of nominal excess
returns denominated in a reference currency is explained by three hedging funds: a nationless
(or logarithmic) world market portfolio; portfolios hedging against deviations from PPP; and
14
portfolios hedging against variations in the investment opportunity set of an international
investor. Consider the II-CAPM in its discrete-time approximate version. The vector wtl of
the l-country (l = 1, ..., L + 1) optimal weights on the risky assets is given by
l
−1 l
l −1 l
wtl = αl Σ−1
rrt [Et (rt+1 ) − rf t 1] + (1 − α )Σrrt srπt + β Σrrt sryk t ,
(32)
where slryk t and slrπt are the vectors of time-varying covariances between each country’s
security returns and the k-th state variable and level of inflation, respectively. Moreover, I
l
JW
t
Wl
WWt t
define αl ≡ − J l
2
and β l ≡ (αl ) ×
l
JW
y
kt
Wtl
.10 Let wτ t denote the N × 1 vector of unscaled
weights of the tangency portfolio
wτ t = Σ−1
rrt [Et (rt+1 ) − rf t 1] .
(33)
l
l
Let wyt
and wπt
denote the N × 1 vectors of unscaled weights hedging against variations in
the investment opportunity set and against deviations from PPP, respectively, so that
l
l
= Σ−1
wyt
rrt sryk t
and
(34)
l
l
wπt
= Σ−1
rrt sryt .
(35)
Then I can rewrite (32) as
l
l
wtl = αl wτ t + (1 − αl )wπt
+ β l wyt
.
PL+1
αl W l
Aggregating over countries and defining αm = Pl=1
L+1
l=1
obtain
m
wmt = α wτ t +
L+1
X
l
απl wπt
l=1
+
Wl
L+1
X
(36)
l
l
l
(1−α )W
W
, αlπ = P
, αyl = PβL+1
L+1
l
l=1
l
αly wyt
.
W
l=1
l
Wl
, I
(37)
l=1
Hence the tangency portfolio, which prices all security returns, is a combination of the global
market portfolio and the portfolios hedging against deviations from PPP and movements
in the investment opportunity set of an international investor. Notice that the II-CAPM
collapses to the IS-CAPM when the investment opportunity set is constant and there are
no deviations from PPP.11 The II-CAPM collapses to the I-CAPM when the investment
10
11
See Appendix B for a derivation of (32).
Note that in this case, using either nominal or real stock returns to test the CAPM returns the same
results.
15
opportunity set is constant or, equivalently, when the weights associated with the hedging
demands are set equal to zero. Moreover, when the inflation rate of country l, expressed in
its home currency, is zero or non stochastic, the L + 1 inflation hedging funds of Adler and
Dumas model collapse to L exchange rate hedging funds:12 the I-CAPM (PPP) collapses
to the I-CAPM (SPOT). The previous setup provides a rationale for using nominal returns
in the analysis. The absence of money illusion also makes it possible to express nominal
returns in a reference currency and, without loss of generality, in excess of a measurement
currency risk-free rate.13 The problem of testing these different versions of the international
CAPM simply reduces to the problem of testing that the tangency portfolio is mean—variance
efficient. The international asset pricing models discussed above can be conveniently stated
in terms of their assumptions on the nominal (normalized) minimum—variance kernel q ? .14
Hence, I define the vector yt+1 as
yt+1 ≡ [ym,t+1 , yf,t+1 , yπ,t+1 , yh,t+1 ]> ,
where ym,t+1 is the rate of return on the world market portfolio, yf,t+1 is the L × 1 vector
of logarithmic changes of the rates of appreciation of the measurement currency, yπ,t+1 is
the (L + 1) × 1 vector of innovations in the inflation rates, and yh,t+1 is the K × 1 vector of
demands hedging against variations in the investment opportunity set, where K represents
the number of economic factors used in the analysis. The corresponding mimicking-portfolio
?
returns, yt+1
, are formed using the methodology described in Section I.D.
Consider, for example, the II-CAPM. The corresponding normalized minimum—variance
12
13
See, for example, Solnik (1974), Sercu (1980), and Grauer, Litzenberger and Stehle (1976).
In the IS-CAPM, translating returns into a new currency and measuring excess returns relative to the
new currency risk-free rate would leave the intercept term equal to zero. In the I-CAPM and II-CAPM,
the new currency foreign exchange premium would be replaced by the old currency exchange risk premium.
Nonetheless, the introduction of conditioning information and the expansion of the set of primitive securities
to include managed portfolios might be affected by the choice of the measurement currency. Hence, the
pricing implications delivered by alternative asset pricing models might differ according to the reference
currency considered. Dumas and Solnik (1995) found that the choice of the measurement currency did not
affect their conclusions in terms of rejection and acceptance of the international CAPM.
14
See Dumas and Solnik (1995) for a similar interpretation of international asset pricing models.
16
?
pricing kernel, qt+1
, can be written as
?
?
?>
?>
qt+1
= 1 − ym,t+1
λmt − yπ,t+1
λπt − yh,t+1
λht ,
(38)
or, when inflation rates in each country are non-random, as
?
?
?>
?>
qt+1
= 1 − ym,t+1
λmt − yf,t+1
λf t − yh,t+1
λht ,
(39)
where λmt , λf t , λπt and λht are commensurable coefficient vectors.
Using (5), I have
?
?>
?>
)λmt + Et (rt+1 yπ,t+1
)λπt + Et (rt+1 yh,t+1
)λht
Et (rt+1 ) − 1rft = Et (rt+1 ym,t+1
(40)
?
?>
?>
Et (rt+1 ) − 1rft = Et (rt+1 ym,t+1
)λmt + Et (rt+1 yf,t+1
)λf t + Et (rt+1 yh,t+1
)λht .
(41)
and
?
?>
?>
?>
), β f t ≡ Et (rt+1 yf,t+1
), β πt ≡ Et (rt+1 yπ,t+1
), and β ht ≡ Et (rt+1 yh,t+1
)
Let β mt ≡ Et (rt+1 ym,t+1
denote the (arrays of) the conditional “betas” associated with the economic variables y. I
can rewrite (40) and (41) to obtain the international intertemporal CAPM linear pricing
results
Et (rt+1 ) − 1rft = β mt λmt + β πt λπt + β ht λht
(42)
Et (rt+1 ) − 1rft = β mt λmt + β ft λft + β ht λht .
(43)
and
Equations (42) and (43) state that the conditional risk premium on any asset is a linear
combination of the conditional risk premia on the different sources of economic and financial
risks. The previous equations collapse to the IS-CAPM when PPP holds and hedging demands are equal to zero while they reduce to the Adler and Dumas I-CAPM when hedging
demands are equal to zero.15
15
Specifically, in testing the II-CAPM, I overparameterize the model and use demands hedging against
inflation and foreign exchange risk simultaneously.
17
II.
Estimation
This section describes the methodology I use in the empirical analysis. First, I illustrate the
approach taken to document time variation in the first and second moments of international
asset returns. Second, I illustrate the estimation of the coefficients of the minimum-variance
kernel. Third, I illustrate the estimation of the economic risk premia using two approaches
based on the minimum-variance kernel and mimicking portfolios. Fourth, I illustrate how
the explicit asset pricing models discussed in the previous section are tested.
A.
Selecting Instruments
Let zt denote a J × 1 vector of instruments whose realizations belong to the information set
at the beginning of each investment period. Without loss of generality, I assume the first
element of zt to be unity, z1t = 1. I model the conditional mean and conditional volatility
of asset returns as linear functions of the instruments. Namely, I assume
Et (rt+1 ) = µr1 + µr2 z2t + . . . + µJ1 zJt , and
Et [|rt+1 − Et (rt+1 )|] = vr1 + vr2 z2t + . . . + vJ1 zJt .
(44)
Hence, the mean and volatility coefficients can be estimated by exactly-identified GMM.
B.
The Minimum-Variance Kernel
I postulate that the coefficients α0t and αt of the minimum-variance pricing kernel are linear
functions of the instruments zt ;
α0t = 1 − Et (rt+1 )> αt = α0 (zt ) ,
(45)
αt = −Σ−1
rrt [Et (rt+1 ) − rf t 1] = α(zt ) ,
(46)
and
where αj , j = 1, . . . , J, are N × 1 coefficient vectors.
18
Note that this approach can be interpreted as the scaling of asset returns with instruments
(or the expansion of the set of securities to include managed portfolios) which is common in
the asset pricing literature.16 In fact, I can write
?
)zt = zt , and
Et (qt+1
?
Et (qt+1
rt+1 ) ⊗ zt = rf t 1 ⊗ zt .
(47)
(48)
Assuming stationarity, and applying the law of iterated expectations, I have
?
E(qt+1
zt ) = E(zt ) , and
?
rt+1 ⊗ zt ) = E(rf t 1 ⊗ zt ) .
E(qt+1
(49)
(50)
?
The two conditions above ensure that, conditioning on zt , qt+1
has mean one and correctly
prices the securities under consideration.
It is convenient at this point to define

and let ιzt ≡ ιt ⊗ zt . Also, I define
ιt ≡ 

1
rf t 1


ra,t+1 ≡ 



1
rt+1
(51)



(52)
and rza,t+1 ≡ ra,t+1 ⊗ zt . Hence, I can rewrite (49) and (50) as
?
E(qt+1
rza,t+1 ) = E(ιzt ) .
(53)
?
z
z
The minimum-variance kernel satisfying (53) has the form qt+1
≡ rz>
a,t+1 αa , where αa is an
(N + 1)J coefficient vector. I have
−1
z
αza = [E(rza,t+1 rz>
a,t+1 )] E(ιt ) .
16
(54)
This scaling procedure has an intuitive interpretation. The scaled returns are the returns on managed
portfolios in which the manager invests more or less according to the signals zt .
19
Hence, the minimum-variance kernel satisfying (53) has the form
?
qt+1
= (αza1 + αza2 z2t + . . . + αzaJ zJt )
z
z2t + . . . αza,2J zJt )r1,t+1
+(αza,J+1 + αa,J+2
+...
z
+(αza,NJ+1 + αa,NJ+2
z2t + . . . αza,J(N+1) zJt )rN,t+1 ,
(55)
which is equivalent to imposing the assumptions (45) and (46). The analysis above still
applies when the positivity constraint is imposed.17 In this case the minimum-variance
z +
kernel has the form q̃t+1 ≡ (rz>
a,t+1 α̃a ) .
Note that when the set of instruments used in the analysis only contains a constant,
this approach can be interpreted as projecting the minimum—variance kernel on the set of
primitive securities.
C.
Economic Risk Premia
In order to estimate the conditional risk premia associated with the variables ykt , I use
two approaches. First, I consider the conditional covariance between the minimum-variance
kernel and yk . Second, I construct mimicking portfolios and I estimate their conditional risk
premia.
In implementing the first approach, I assume
?
?
λ?kt ≡ −Covt (qt+1
, yk,t+1 ) = −Et [(qt+1
− 1)yk,t+1 ] = λk1 + λk2 z2t + . . . + λkJ zJt . (56)
Without loss of generality, I also assume Var(zjt ) = 1. Hence the coefficients λkj can be
interpreted as the change in the conditional risk premium for a one-standard-deviation change
in the instrument. The assumption that the conditional risk premia are determined by the
set of instruments zt is quite natural: the conditional risk premia assigned by the minimumvariance kernel are the mean cash flows generated by the hedging portfolio financed at the
17
Appendix C shows how to estimate economic risk premia under the no arbitrage condition of non-
negativity of the normalized pricing kernel.
20
riskless rate. Hence, if the variables in zt are predictors of asset returns, they should also
predict excess returns on the hedging portfolios. In fact, modeling the time-variation of risk
premia in this fashion is common to other studies (Ferson and Harvey (1991), for example).
Note that, without loss of generality, I can assume E(zjt ) = 0, i = 2, . . . , J. This means
that
E(λ?kt ) = λk1 .
(57)
The distinction between conditional and unconditional risk premia is important because,
even if the unconditional premium is close to zero, the conditional premia may take values
over time which are significantly different from zero.
When the positivity restriction is imposed, I have
λ̃kt ≡ −Covt (q̃t+1 , yk,t+1 ) = −Et [(q̃t+1 − 1)yk,t+1 ] = λk1 + λk2 z2t + . . . + λkJ zJt .
(58)
The second approach to the estimation of the economic risk premia is based on the
construction of hedging portfolios. I postulate that the coefficients α0yk t and αyk t of the
?
hedging portfolio are linear functions of the instruments zt . Hence, I have yk,t+1
= αzayk rza,t+1 ,
where
−1
z
αzayk = [E(rza,t+1 rz>
a,t+1 )] E(ra,t+1 yk,t+1 ) .
(59)
As with the coefficients of the minimum-variance kernel, one can show that this approach
is equivalent to projecting the risk variables on both the returns on the primitive securities
and the cash flows of dynamic strategies. In particular, when zt = z1t = 1 this approach is
equivalent to projecting the risk variables only on the set of primitive securities.
Finally, I estimate the Sharpe ratios of the hedging portfolios, since they might differ
substantially from the conditional risk premia (see discussion above). Hence, I take the ratio
between the risk premium and the volatility of the hedging portfolios cash flows.
21
D.
Tests of International Asset-Pricing Models
Each of the international asset-pricing models I considered has specific pricing implications,
as previously discussed. For convenience, I discuss my tests with respect to the implications
of the static international capital asset pricing model (IS-CAPM). The approach to test the
other models is analogous.
Recall that the IS-CAPM implies that the portfolio hedging global market risk prices all
securities. Using a standard GMM test I test whether a pricing kernel linear in the cash
?
?
?
flows of this portfolio, qm,t+1
, prices all securities. I have qm,t+1
= yma,t+1
αzma , where
?
?>
αzma = E(yma,t+1
yma,t+1
)−1 E(ιmt )
(60)
and


zt
?
≡ 
yma,t+1
z
rz>
a,t+1 αaym




ιmt ≡ 
zt
z
ιz>
t αaym


(61)
.
(62)


I test the moment conditions
?
E(qm,t+1
rza,t+1 ) = E(ιzt+1 ) .
(63)
This corresponds to the test of overidentifying restrictions pioneered by Hansen and Singleton
(1982). I have J +K coefficients in the vector αzma and J +N J moment conditions, for a total
of N J − K overidentifying restrictions. If the test failed, this would mean that the pricing
errors generated by the IS-CAPM are statistically significant, and the model is rejected.
?
Second, I compare the standard deviation of qm,t+1
to the standard deviation of the
?
minimum-variance normalized kernel, qt+1
. Namely, I estimate the difference
q
?
Var(qm,t+1
)−
q
?
Var(qt+1
).
(64)
This corresponds to the HJV test. Since the variances of the two kernels correspond to
the average squared Sharpe ratios of the global market-hedging portfolio and the tangency
22
portfolio, this test is equivalent to a test of the mean-variance efficiency of the world markethedging portfolio.
?
?
Third, I calculate the standard deviation of the difference between qm,t+1
and qt+1
,
q
?
?
Var(qt+1
− qm,t+1
).
(65)
This corresponds to the HJD test.18 If the test failed, this would mean that the difference between the pricing kernel generated by the IS-CAPM and any admissible kernel is statistically
significant, and the model is rejected.
III.
Data
This section illustrates the data used in the empirical analysis. The period considered is
April 1970 through October 1998 for stock returns and economic variables and March 1970
through September 1998 for instrumental variables. Data are monthly. The starting and
ending dates for the sample are dictated by macroeconomic and financial data availability.
A.
Asset Returns
I use the Morgan Stanley Capital International (MSCI) national equity indices. The nominal
returns are denominated in U.S. dollars and are calculated with dividends. All indices have
a common basis of 100 in December 1969. The indices are constructed using the Laspeyres
method, which approximates value weighting.19 U.S. dollar returns are calculated by using
the closing European interbank currency rates from MSCI. I choose the four countries with
the largest market capitalization: United States; United Kingdom; Japan; and Germany.20
Table I shows summary statistics for monthly returns on stock indices from MSCI.
18
?
?
?
Since I allow a constant in qm,t+1
, E(qm,t+1
) = E(qt+1
) = 1. Hence,
q
?
?
Hansen-Jagannathan distance, equals Var(qt+1
− qm,t+1
).
19
20
q
?
?
E(qt+1
− qm,t+1
)2 , which is the
See MSCI Methodology & Index Policy for a detailed description of MSCI’s indices and properties.
As of 1996, the market capitalization weight for these countries is 76.2% of the market capitalization
world.
23
B.
Economic Variables and Instruments
I use inflation rates, spot exchange rates, and the rate of return on the world market portfolio
as the relevant sources of risk in this study. Consumer price indices are from International
Financial Statistics (IFS) and are denominated in local currency. Spot exchange rates are
from MSCI. The world equity market index is a value—weighted combination of the country
returns tracked by MSCI. I proxy foreign exchange risk with logarithmic changes in the
spot exchange rates (SPOT); inflation risk with an ARIMA(0,1,1) for inflation (INFL); and
global market risk with the level of world equity returns (WLDMK). Note that the variable
INFL represents unexpected inflation and that the variable SPOT represents innovations
in the exchange rate if I assume that spot rates follow a random walk. Table II contains
summary statistics for the relevant risk variables. In choosing the set of instruments, I
concentrate on a set of variables which have been previously used in tests of multiple-beta
models and/or in studies of stock-return predictability.21 These variables are statistically
significant in multivariate predictive regressions of means and volatilities and/or they have
special economic significance. The instruments include a constant, a January dummy, and
the following five variables:
DINFLUS is the lagged difference in the U.S. monthly rate of inflation (IFS).
EURO represents the one—month Eurodollar deposit rate (DRI) and performs as the
conditionally nominal risk—free asset in the analysis.
USDIVYLD denotes the U.S. monthly dividend yield (MSCI) in excess of the 1-month
Eurodollar deposit rate. Specifically, the monthly dividend yield is equal to 1/12 of
the ratio between the previous year dividend and the index at the end of each month.
WLDMK denotes the lagged value of the world stock market monthly returns (MSCI).
DEFPREM denotes the U.S. default premium as given by the return difference between
Moody’s Baa-rated and Aaa-rated bonds (SBBI Yearbook).
21
See, for example, Ferson and Harvey (1993), Dumas and Solnik (1995), and De Santis and Gérard (1998).
24
I select as instruments the previous variables as a proxy for the information investors use to
set prices in the market.
Table III presents summary statistics of the instrumental variables.
IV.
Selecting Instruments
In this section I select a set of instruments by performing an analysis of predictability. Specifically, I look at the ability of some variables used in previous studies of international asset
pricing to predict variation in the first and second conditional moments. This preliminary
analysis is relevant for three reasons. First, it identifies the information set of an international investor. Second, it makes it possible to study the patterns of time variation in the
conditional risk premia. Third, it provides a rationale for including a third hedging portfolio
in alternative international CAPM specifications besides portfolios hedging against global
market and inflation (or currency) risks.
Table IV presents my results. I report the coefficients of the mean and variance equations,
as well as three statistics: µ̄r is the average slope coefficient in the mean equations; v̄r is the
average slope coefficient in the variance equations; and µ̄r − v̄r is the difference between the
two average slope coefficients. These statistics provide an indication of the net effect of the
instruments on the investment opportunity set.
The following patterns emerge from this analysis (see especially Panel C):
The lagged change in the U.S. inflation rate (DINFLUS) has a negative and significant
average impact on returns and a negative, but insignificant, average impact on return
volatility. The net effect on the investment-opportunity set is strongly negative.
The Eurodollar deposit rate (EURO) has a positive and significant average impact on
returns and a negative and significant average impact on return volatility. The net
effect is strongly positive.
The U.S. dividend yield in excess of the EURO (USDIVYLD) has a positive and
25
significant average effect on returns and a negative and significant overall effect on
volatility. The net effect is positive.
The lagged world market equity return (WLDMK) has an overall positive, but insignificant, impact on returns. The impact on volatility is negative and partially significant.
The net effect is positive but not large.
The U.S. default premium (DEFPREM) positively affects returns and is partially significant. The overall effect on volatility is negative and partially significant. The net
effect is positive.
In summary, I can rank the net effects of the different variables on the investmentopportunity set as follows (from largest to smallest): USDIVYLD, EURO, DEFPREM,
WLDMK, DINFLUS.22
V.
Risk Premia and Sharpe Ratios
In this section I report estimates of the risk premia associated with the economic variables
and look at their patterns of time variation.23 I use the instrumental variables selected in
the previous section to document the patterns of time variation of the conditional premia.
Table V reports estimates of the coefficients of the economic risk premia estimated using
?
. Since the instruments are demeaned, the intercept term
the minimum-variance kernel qt+1
can be interpreted as the unconditional risk premium on yk,t+1 . I do not report coefficient
estimates of the economic risk premia estimated using the non-negative minimum-variance
kernel q̃t+1 because there is no substantial difference with respect to the estimates reported
in Table VI. Table VII reports coefficient estimates of the economic risk premia estimated
using the hedging portfolios.
22
23
Note that Germany, among all countries in the study, exhibits the weakest patterns of predictability.
Note that the instruments do not coincide with the past values of the economic variables included in the
analysis, with the exception of the world market portfolio.
26
As noted in section I.D. above, the expected excess cash flows on the hedging portfolios
?
coincide with risk premia assigned by qt+1
. Yet, the realized excess cash flows on the hedging
?
portfolios in general differ from (qt+1
− 1)yk,t+1 . Hence, the estimates of the unconditional
risk premia using the “q ? ” and the “hedging portfolio” approaches will coincide, although
their standard errors may differ. In addition, the impact of the conditioning variables on the
conditional risk premia will also differ. Note that when y ? and q ? are mapped onto the space
spanned by the primitive securities and are estimated inside of the algorithm, the standard
errors of the unconditional risk premia coincide (see Panel A of Tables V and VI).
The tables report two sets of t-ratios. The first t-ratio is obtained using a two-step
?
?
procedure: I first estimate the coefficients of qt+1
and yk,t+1
; I then estimate the risk pre-
mia by exactly-identified GMM algorithm. The second t-ratio is obtained estimating all
parameters inside the GMM algorithm. The reason for the two separate approaches is that
I am concerned with the large number of estimated parameters when the coefficients of the
minimum-variance kernels and the hedging portfolios are estimated by GMM. As it turns
out, the t-ratios change only marginally across the two procedures. In all tests, standard
errors are adjusted for heteroskedasticity and serial correlation.
The results are also very similar across estimation methods. The main difference is that
the estimation based on the q ? tends, in most of the cases, to yield tighter standard errors.
The following patterns emerge from the tables:
The unconditional inflation risk premia are negative (except for Germany) but not significant. When managed portfolios are ruled out from the analysis, only U.S. inflation
seems to be priced and to be significant across estimation techniques. No significant
pattern of time variation emerges from the analysis of the foreign inflation conditional
premia.
The unconditional foreign exchange risk premia are negative. Even if I am not using
any equilibrium model to compute economic risk premia, this result is consistent with
the predictions of Adler’s and Dumas’ international CAPM. They show that if the
degree of risk aversion of investors is greater than one, foreign exchange risk premia
27
should be negative. Nonetheless, the unconditional estimates of currency risk premia
are not statistically significant. This result also supports Adler and Dumas (1995)
and De Santis and Gerard (1998). I also find time variation in the estimates of the
conditional premia statistically significant.
The unconditional global market risk premium is positive and statistically significant
across estimation techniques. This result seems to be robust and does not support the
analysis of Adler and Dumas (1995) or De Santis and Gérard (1998). They actually
find that the world market unconditional risk premium is positive but insignificant. On
the other hand, my findings are supported by the work of Hodrick, Ng, and Sengmüller
(1999). Moreover, I find significant time variation in the global market conditional risk
premium.
In addition, I estimated the unconditional and conditional premia implied by the set
of economic and financial factors. Estimation results (not reported in the paper) do not
provide any evidence of statistical significance of these premia. Even if the factors exhibit
some non—trivial patterns of predictability for the first and second moment of asset returns,
they do not seem to be priced.
In summary, I find that the signs of the risk premia associated with foreign exchange
and inflation risks are largely consistent with the theoretical predictions of several models of
international asset pricing. At the same time, foreign inflation unconditional and conditional
risk premia and foreign exchange unconditional risk premia are imprecisely estimated. On
the contrary, I document significant patterns of time variation of foreign exchange premia
and find that constant and time-varying global market risk premia are precisely estimated.
The remaining economic and financial factors are not priced, both unconditionally and conditionally.
The risk premia estimated above coincide with the Sharpe ratios of exact mimicking portfolios. But, in general, economic factors can be tracked only imperfectly by asset returns.
Hence, in order to obtain Sharpe ratios on traded portfolios I need to standardize the estimates obtained above by the volatility of the approximate mimicking portfolio returns. The
28
composition of the mimicking portfolios is estimated separately from the Sharpe ratios.
Table VII presents the Sharpe ratios on the eight hedging portfolios. I find that the
volatilities of the mimicking portfolios cash flows are strongly significant. On the other
hand, given the statistical insignificance of the mean estimates, the Sharpe ratios on the
hedging portfolios are insignificant, with the exception of the global market portfolio.
VI.
Hedging Demands and Tests of International Asset
Pricing Models
In this section I discuss the results of tests of four explicit asset pricing models: IS-CAPM,
I-CAPM (PPP), I-CAPM (SPOT), and II-CAPM.24 Results of the tests are presented in
Table VIII and in Table IX. The tests are performed using the full set of instruments zt
(“With conditioning information”).
In Table VIII, I report the χ2 statistic associated with a test of the overidentifying
restrictions, the difference between the standard deviation of the candidate pricing kernel
?
and the standard deviation of qt+1
, the HJV statistic, and the standard deviation of the
?
difference between the candidate pricing kernel and qt+1
, the HJD statistic. I also report
the p-values associated with the χ2 test, and the t-ratios associated with the HJV and HJD
statistics.
In the test of overidentifying restrictions, the coefficients of the candidate kernel are
estimated by GMM, although the composition of the mimicking portfolios is estimated separately, outside of the GMM algorithm. In the other two tests, the coefficients of the candidate
kernel are estimated separately.
I find that the χ2 tests do not reject all five models conditionally. In particular, the
24
The additional mimicking portfolios considered in the II-CAPM specification are obtained via an OLS
regression of DINFLUS, EURO, USDIVYLD and DEFPREM on the augmented span of managed portfolio
returns.
29
II-CAPM(PPP) can be weakly rejected at the 5% significance level but not at the 1% level.
On the contrary, tests of the HJ bounds and of the HJ distance measure indicate that
all the models are rejected by the data — the standard deviation of the candidate kernels
?
. Nonetheless, as shown in Table IX, there
is always substantially lower than that of qt+1
are differences in performance. The IS-CAPM generates the least volatile pricing kernel.
The I-CAPM (SPOT) of Grauer, Litzenberger, and Stehle (1976) generates pricing kernels
with somewhat higher volatility. The second most volatile pricing kernel is generated by the
Adler and Dumas model. The most volatile kernel is generated by the II-CAPM (PPP).
The difference between the volatility of the II-CAPM (PPP) and I-CAPM (SPOT) kernels
is substantial (68 percent) while the difference between the volatility of the II-CAPM (PPP)
and I-CAPM (PPP) kernels is close to 16 percent. Note that the standard deviation of a
pricing kernel coincides with the average Sharpe ratio of the tangency portfolio constructed
using the underlying set of assets. Hence, the HJV statistic can be interpreted as the
difference between two average Sharpe ratios.
Overall, the evidence from these tests is that while all five models are not rejected by
the Hansen and Singleton (1982) χ2 test of overidentifying restrictions, the same models are
formally rejected by the tests of HJ variance bounds and HJ distance. These results are in
sharp contrast with the findings of Dumas and Solnik (1995) as well as De Santis and Gérard
(1998). These authors do not reject the international CAPM in its conditional version on the
basis of tests of overidentifying restrictions and cross-equations restrictions. The use of test
statistics that do not reward the variability of alternative admissible pricing kernels allows
me to reject the conditional international CAPM in all its specifications.
I also investigate the size and the significance of the unconditional demands induced by
hedging against global market, inflation, and foreign exchange risks. The estimates of these
hedging demands correspond to the coefficients of the mimicking portfolios that appear in
the kernel specification described in Section II.D. The hedging demands are normalized to
sum up to one. Standard errors are computed using the delta method.
As shown in Table X, the scaled hedging demands are precisely estimated. In particular,
30
the coefficients associated with the cash-flows of the inflation hedging portfolios are large in
magnitude and statistically significant. This result further explains why the international
intertemporal CAPM in presence of deviations from PPP has better pricing implications
than the other asset pricing models.25
VII.
Conclusions
This paper presents a new approach for the estimation of risk premia associated with observable sources of risk, which is based on the moments of the minimum-variance kernel of
Hansen and Jagannathan (1991). I also provide extensive evidence on the performance of
four explicit asset pricing models: the IS-CAPM; the I-CAPM (PPP); the I-CAPM (SPOT);
and the II-CAPM in presence of deviations from PPP.
In sharp contrast with Dumas and Solnik (1995) and De Santis and Gérard (1998), but
in line with Hodrick, Ng, and Sengmüller (1999), I find the global market risk is priced both
conditionally and unconditionally.
All international asset-pricing models are formally rejected by the data when the testing
methodology is stringent enough. In addition, the II-CAPM (PPP) that I construct and test
using mimicking portfolios outperforms the IS-CAPM. It generates a pricing kernel which is
72% more volatile than the one of the IS-CAPM. For these models, the differences is Hansen
& Jagannathan variance bounds and distance measures are large and statistically significant.
Hence, I add to the existing literature showing that, introducing deviations from PPP and
dynamic hedging, the II-CAPM (PPP) is able to generate more accurate pricing implications
than the other versions of the international CAPM. My findings show that the II-CAPM
(PPP) outperforms the competitive international asset-pricing models because most of the
economic and financial factors considered significantly affect the first and second conditional
moments of asset returns. Finally, the result that the international CAPM in its alternative
25
The inflation-mimicking portfolios make the II-CAPM(PPP) kernel more volatile and hence closer to
the bounds of the minimum-variance kernel.
31
specifications does not hold is consistent with the evidence of at least mild segmentation of
international equity markets and is supported by Hodrick, Ng, and Sengmüller (1999).
Future work should investigate the sensitivity of these results to the level of aggregation
of asset returns, to the choice of the investment opportunity set of an investor, and to the
measurement currency used in the analysis.
32
Appendix A
A.
Unobservable Factors: Principal Components
One common approach to the study of asset returns is based on principal-component analysis.
Let St denote the matrix whose columns are the orthonormal eigenvectors of Σrrt . Since
S>
t St = I, I have
>
rt+1 = S>
t St rt+1 ≡ St rp,t+1 ,
(66)
where rp,t+1 ≡ St rt+1 is the vector of orthogonal factor-portfolio returns.26 This makes it
?
possible to rewrite the kernel qt+1
as
?
−1 >
qt+1
= 1 − [rt+1 − Et (rt+1 )]> S>
t St Σrrt St St [Et (rt+1 ) − rf t 1]
= 1 − [rp,t+1 − Et (rp,t+1 )]> Vt−1 [Et (rp,t+1 ) − rf t S>
t 1] ,
(67)
where Vt is a diagonal matrix whose viit element is the i-th eigenvalue of Σrrt .
Three main insights can be developed based on the expressions above. First, the minimumvariance kernel is a linear combination of the factor-portfolio returns
?
qt+1
=1−
N
X
rpi,t+1
i=1
− Et (rpi,t+1 )
Spit ,
√
viit
(68)
where Spit is the Sharpe ratio on the i-th factor-portfolio. Hence, the (standardized) risk
premia associated with the factor portfolios are the coefficients relating the (standardized)
innovations in the factor portfolio returns to the minimum-variance kernel. Moreover, equation (68) highlights how the variance of the minimum-variance kernel, and hence the squared
Sharpe ratio of the tangency portfolio, can be decomposed according to the squared Sharpe
ratios of the factor portfolios
?
Vart (qt+1
) = Sτ2t =
N
X
2
Spit
.
(69)
i=1
26
Note that the weights of the factor portfolios do not sum to one: the eigenvectors are normalized so that
the sum of the squared elements is one.
33
Second, since the factor portfolio returns are orthogonal to each other, the covariance
?
between qt+1
and any risk variable yk,t+1 can be written as a linear combination of the sum
of the covariances between each of the factor portfolio returns and yk,t+1 :
λkt = −
N
X
Covt (rpi,t+1 , yk,t+1 )
Spit .
√
viit
i=1
(70)
This expression allows for a breakdown of the risk premium on an observable risk variable
into the components due to the different factor portfolios.
Third, the ability of a subset of the factor portfolios to price all assets can be tested in
the same way as any candidate pricing kernel.
B.
Observable Factors: Multi-beta Models
A multi-beta model implies that expected excess returns are linear in the sensitivities of the
returns to the risk variables, with coefficients given by the risk premia associated with the
factors:
Et (rt+1 ) − rf t 1 = β t λt ,
(71)
where β t is an N × K matrix of projection coefficients of the returns on the risk variables.
Hence, excess returns are described by the model
rt+1 − rf t 1 = β t λt + β t yt+1 + et+1 ,
(72)
where et+1 is a vector of N × 1 mean-zero perturbances orthogonal to the risk variables yt+1 ,
with covariance matrix Σeet .
?
Consider now the risk premia assigned by the minimum-variance kernel qt+1
. From (16)
I have
λ?t = αyt β t λt
−1
= Σyrt Σ−1
rrt Σryt Σyyt λt .
34
(73)
In the special case where Σyyt = I, i.e., the factors are orthogonal with unit variance, the
equation above simplifies to
λ?t = Σyrt Σ−1
rrt Σryt λt .
(74)
It is straightforward to verify that the Σyrt Σ−1
rrt Σryt matrix equals the covariance matrix of the
projections of the risk variables onto the span of asset returns, Σy? y? t ,27 i.e., λ?t = Σy? y? t λt .
?
Hence, the risk premia assigned by qt+1
are linear combinations of the multi-beta premia
through coefficients, which depend on the covariance matrix of the hedging-portfolio cash
flows. In the two-factor case, for example,
λ?1t = σy2? 1t λ1t + σy? 12t λ2t ,
(75)
2
where σy1t
is also the R2 of the projection of the first risk variable on the span of returns. In
the special case where y1t is perfectly tracked by the hedging portfolio, σy2? 1t = 1, σy? 12t = 0,
and λ?1t = λ1t .
Appendix B
Equation (32) combines Merton’s (1973) intertemporal CAPM with Adler and Dumas (1983)
international asset pricing model in presence of deviations from PPP. The continuous-time
portfolio selection problem of a representative investor can be stated as follows:28
Max E
Z T
V (C, P, s) ds ,
(76)
t
where C = C(W, P, yk , t) denotes nominal consumption expenditures, P is the price level
index, V is a function homogeneous of degree zero in C and P expressing the instantaneous rate of indirect utility, and yk is a state variable that affects utility through nominal
consumption. Following Merton (1969), the wealth dynamics can be written as
N
X
dW = [
i=1
wi (µi − rf ) + rf ]W dt − Cdt −
N
X
wi σi dzi ,
(77)
i=1
27
−1
−1
?
I have Vart (yt+1
) = Vart (αyt rt+1 ) = Σyrt Σ−1
rrt Σrrt Σrrt Σryt = Σyrt Σrrt Σryt .
28
See Appendix in Adler and Dumas (1983) for a detailed explanation of the necessary assumptions.
35
where w = {wi } is (N + 1) × 1 vector of weights, µi is the instantaneous expected nominal
rate of return on security i expressed in a reference currency, σi is the instantaneous standard
deviation of the nominal rate of return on security i, rf is the risk-free rate expressed in a
reference currency, and dzi is the white noise of a standard Wiener process. Denoting with
J(W, P, yk , t) the maximum value of (76) subject to (77), the Bellman principle states that
total expected rate of increase of this function must be identically zero, so that
N
X
Max
0 =
{V (C, P, yk , t) + Jt + JW [−C + W ( wi (µi − rf ) + rf )] + Jyk α
(C, w)
i=0
N
N X
N
X
X
1
1
1
+JP P π + JW W W 2
wi wj σij + Jyy s2 + JP P σπ2 P 2 + JW P W P
wi σiπ
2
2
2
i=1 j=1
i=1
+Jyk W W
N
X
i=1
wi σiyk + Jyk P σyk π } ,
(78)
where π is the inflation rate in each country expressed in local units, σij are the instantaneous
covariances of the nominal rates of return on the various securities, σπ2 is the instantaneous
variance of the inflation rate, α is the mean value of the state variable yk , σiπ is the covariance between security i and the inflation rate π, and σyk π is the covariance between the
state variable yk and the inflation rate π.29 Moreover, the homogeneity of degree zero of
the function V implies that J(W, P, yk , t) and C(W, P, yk , t) that satisfy (78) must be homogeneous of degree zero in W and P : JP ≡ −(W/P )JW , JP W ≡ (−1/P )JW − (W/P )JW W ,
JP P = 2(W/P 2 )JW + (W/P )2 JW W . Hence, (78) can be rewritten as
0 =
N
X
Max
{V (C, P, yk , t) + Jt + JW [−C + W ( wi (µi − rf ) + rf )] + Jyk α − W πJW
(C, w)
i=0
N
N X
N
X
X
1
1
1
+ JW W W 2
wi wj σij + Jyy s2 + W JW σπ2 + σπ2 W 2 JW W − JW W
wi σiπ
2
2
2
i=1 j=1
i=1
−W 2 JW W
N
X
wi σiπ + Jyk W W
i=1
N
X
i=1
wi σiyk + Jyk P σyk π } .
(79)
Taking the first order conditions of (79) with respect to C and w, I obtain
VC = JW , and
0 = JW (µi − rf ) + W JW W
29
(80)
N
X
j=1
wj σij − JW σiπ − W JW W σiπ + Jyk W σiyk .
See Chapter 13 of Ingersoll (1987) for a definition of the dynamics of the state variables.
36
(81)
Solving (79) for the optimal portfolio of risky assets of investor l directly in vector notation
and reintroducing the time dimension, I obtain (32)
l
−1 l
l −1 l
wtl = αl Σ−1
rrt [Et (rt+1 ) − rf t 1] + (1 − α )Σrrt srπt + β Σrrt sryk t ,
(82)
where slryk t and slrπt are the vectors of time-varying covariances between each country’s
security returns and the k-th state variable and level of inflation respectively. Moreover, I
l
JW
t
l
W W t Wt
define αl ≡ − J l
2
and β l ≡ (αl ) ×
l
JW
y
kt
Wtl
.
Appendix C
?
While the minimum-variance pricing kernel, qt+1
, satisfies the law of one price (equation
?
> 0. Nonetheless, as in
(1)), in general it does not satisfy the no-arbitrage condition, qt+1
HJ, I can extend the analysis to take this restriction into account.
Let α̃t denote an N × 1 coefficient vector, and define q̃t+1 ≡ [1 − (rt+1 − Et (rt+1 ))> α̃t ]+ ≡
max{1 − (rt+1 − Et (rt+1 ))> α̃t , 0}. Assume
Et (q̃t+1 rt+1 ) = rf t 1 .
(83)
The random variable q̃t+1 has the smallest variance among all nonnegative random variables
satisfying restriction (83).
Consider the risk premium λ̃kt assigned by q̃. I can write
λ̃kt ≡ −Et [(q̃t+1 − 1)yk,t+1 ]
?
?
= −Et [(q̃t+1 − 1)yk,t+1
] − Et [(q̃t+1 − 1)(yk,t+1 − yk,t+1
)]λ̃kt
?
= λ?kt − Et [(q̃t+1 − 1)(yk,t+1 − yk,t+1
)] ,
(84)
?
where, in general, Et [(q̃t+1 − 1)(yk,t+1 − yk,t+1
)] 6= 0. Hence, when the positivity restriction is
imposed, the risk premium assigned by the minimum-variance kernel differs from the mean
?
cash flow generated by the hedging portfolio by the quantity −Et [(q̃t+1 − 1)(yk,t+1 − yk,t+1
)].
If q̃t+1 is volatile, and if yk,t+1 is mimicked poorly by its nearest hedge, then there is the
potential for the discrepancy to be substantial.
37
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40
Table I
Summary Statistics of Asset Returns
I report summary statistics of monthly returns on the equity indices of four countries from MSCI. Indices include dividends.
The sample covers the period April 1970 through October 1998 (343 observations). All returns are in percentage points per
month and are denominated in U.S. dollars. “Corrτ ” denotes the autocorrelation coefficient of order τ . “Q36 ” denotes the
Ljung—Box Q statistics of order 36 (p—values in parenthesis).
Panel A: Means, Standard Deviations, and Autocorrelations
Country
United States
United Kingdom
Japan
Germany
Mean Std. Dev.
1.1113
4.4171
1.3304
7.0525
1.2296
6.6757
1.2099
5.9069
Corr1
0.002
0.085
0.091
-0.017
Corr2 Corr3 Corr4 Corr12 Corr24 Corr36
-0.034 0.007 -0.018 0.048 0.005 -0.033
-0.101 0.052 0.013 -0.023 0.047 -0.026
-0.022 0.078 0.043 0.032 0.006 0.033
-0.013 0.052 0.067 -0.017 0.022 0.029
Panel B: Correlation Matrix
Country
United States
United Kingdom
Japan
Germany
United States United Kingdom Japan Germany
1.000
0.505
0.264
0.367
1.000
0.358
0.425
1.000
0.366
1.000
41
Q36
27.586
(0.842)
48.085
(0.086)
46.582
(0.111)
41.417
(0.246)
Table II
Summary Statistics of Economic Variables
I report summary statistics of economic risk factors for the United States, the United Kingdom, Japan and Germany. The sample covers the period
April 1970 through October 1998 (343 observations). IN F L denotes the unexpected rate of inflation (percentage points per month). SP OT denotes
the change in the logarithm of the spot exchange rate (percentage points per month). W LDM K denotes the rate of return on the world market index
from MSCI (percentage points per month). “Corrτ ” denotes the autocorrelation coefficient of order τ . “Q36 ” denotes the Ljung—Box Q statistics of
order 36 (p—values in parenthesis).
Panel A: Means, Standard Deviations, and Autocorrelations
Variable
INFLU S
INFLU K
INFLJAP
INFLGER
SPOT$/GBP
SPOT$/Y EN
SPOT$/DM
WLDMK
Mean
-0.0002
-0.0001
0.0004
-0.0007
0.1049
-0.3288
-0.2312
1.0551
Std. Dev.
0.2406
0.6130
0.6770
0.3129
3.0593
3.3611
3.2740
4.1392
Corr1
0.160
0.164
0.100
0.188
0.086
0.080
0.036
0.068
Corr2
-0.023
-0.019
-0.220
0.051
0.025
0.033
0.070
-0.055
Corr3
-0.112
-0.061
-0.060
-0.019
-0.019
0.048
0.014
0.013
Corr4
-0.154
-0.119
-0.013
-0.082
0.012
0.039
-0.005
-0.021
Corr12
0.167
0.475
0.439
0.299
-0.011
0.073
-0.007
0.053
Corr24
0.093
0.418
0.397
0.094
-0.038
-0.045
0.004
0.045
Corr36
0.138
0.387
0.378
0.165
-0.023
-0.085
0.017
-0.019
Q36
95.189
(0.000)
295.03
(0.000)
373.09
(0.000)
134.34
(0.000)
33.924
(0.568)
37.342
(0.407)
32.180
(0.651)
45.179
(0.140)
Panel B: Correlation Matrix
Variable
INFLU S
INFLU K
INFLJAP
INFLGER
SPOT$/GBP
SPOT$/Y EN
SPOT$/DM
WLDMK
INFLU S
1.000
INFLU K
0.057
1.000
INFLJAP
0.141
0.337
1.000
INFLGER
0.157
0.224
0.082
1.000
42
SPOT$/GBP
0.016
-0.089
-0.009
0.033
1.000
SPOT$/Y EN
0.013
-0.015
0.029
0.082
0.472
1.000
SPOT$/DM
0.095
-0.081
-0.014
0.097
0.650
0.578
1.000
WLDMK
-0.196
0.110
-0.008
0.019
-0.261
-0.310
-0.228
1.000
Table III
Summary Statistics of Instrumental Variables
I report summary statistics of the intruments used in the analysis. The sample covers the period March 1970 through September 1998 (343 observations). DIN F LU S denotes the lagged change in the U.S. rate of inflation. EU RO denotes the Eurodollar deposit rate. U SDIV Y LD denotes the
U.S. dividend yield in excess of the Eurodollar rate. W LDM K denotes the world rate of return. DEF P REM represents the U.S. default premium.
All variables are expressed in percentage points per month. “Corrτ ” denotes the autocorrelation coefficient of order τ . “Q36 ” denotes the Ljung—Box
Q statistics of order 36 (p—values in parenthesis).
Panel A: Means, Standard Deviations, and Autocorrelations
Variable
DINFLUS
EURO
USDIVYLD
WLDMK
DEFPREM
Mean
-0.0012
0.6503
-0.3353
1.1672
0.0141
Std. Dev.
0.2719
0.2780
0.2172
4.1166
1.1768
Corr1
-0.355
0.967
0.950
0.068
-0.198
Corr2
-0.054
0.920
0.880
-0.035
-0.036
Corr3
-0.033
0.879
0.822
0.015
-0.005
Corr4
-0.077
0.844
0.771
-0.024
0.013
Corr12
0.142
0.667
0.543
0.062
-0.067
Corr24
0.157
0.343
0.150
0.046
-0.054
Corr36
0.169
0.151
-0.088
-0.016
0.028
Panel B: Correlation Matrix
Variable
DINFLUS
EURO
USDIVYLD
WLDMK
DEFPREM
DINFLUS
1.000
EURO
-0.008
1.000
USDIVYLD
0.000
-0.952
1.000
43
WLDMK
-0.075
-0.126
0.131
1.000
DEFPREM
0.098
-0.046
0.077
-0.003
1.000
Q36
140.350
(0.000)
4051.0
(0.000)
2768.6
(0.000)
46.196
(0.119)
65.046
(0.002)
Table IV
Instruments Selection and Heteroskedasticity
T-statistics, in parentheses, are adjusted for heteroskedasticity and serial correlation.
Panel A: Slope Estimates of Mean Equations
Var
United States
United Kingdom
Japan
Germany
CONST
JAN
DINFLUS
EURO
USDIVYLD
WLDMK
DEFPREM
0.971
(4.099)
1.076
(2.976)
1.193
(3.236)
1.243
(3.781)
1.716
(1.696)
3.114
(1.393)
0.448
(0.371)
−0.410
(−0.373)
−0.408
(−1.759)
−0.469
(−1.294)
−0.772
(−2.229)
−0.659
(−2.051)
1.266
(1.453)
2.828
(2.468)
2.273
(2.137)
0.457
(0.476)
1.605
(1.893)
3.408
(2.964)
2.835
(2.677)
0.863
(0.898)
−0.101
(−0.349)
0.086
(0.194)
0.629
(1.508)
0.161
(0.434)
0.231
(0.882)
0.084
(0.150)
0.154
(0.392)
0.574
(1.475)
Panel B: Slope Estimates of Variance Equations
Var
United States
United Kingdom
Japan
Germany
CONST
JAN
DINFLUS
EURO
USDIVYLD
WLDMK
DEFPREM
3.192
(21.212)
4.809
(20.170)
5.083
(22.131)
4.458
(21.111)
1.062
(1.739)
1.521
(0.847)
−0.528
(−0.709)
−0.252
(−0.351)
0.105
(0.840)
0.098
(0.427)
−0.364
(−1.595)
0.048
(0.238)
0.039
(0.070)
1.587
(1.848)
−1.262
(−1.956)
−0.825
(−1.361)
−0.224
(−0.414)
1.362
(1.578)
−1.194
(−1.867)
−1.028
(−1.712)
−0.472
(−2.739)
−0.079
(−0.256)
−0.016
(−0.057)
−0.216
(−0.932)
−0.359
(−2.438)
−0.375
(−0.887)
−0.056
(−0.221)
0.106
(0.423)
Panel C: Average Slope Estimates and Differences in Slope Estimates
Var
µ̄r
v̄r
µ̄r − v̄r
DINFLUS
EURO
USDIVYLD
WLDMK
DEFPREM
−0.577
(−2.441)
−0.028
(−0.222)
−0.549
(−1.972)
1.706
(2.360)
−0.115
(0.263)
1.821
(2.095)
2.178
(3.051)
−0.271
(−0.630)
2.449
(2.855)
0.194
(0.712)
−0.196
(−1.181)
0.390
(1.361)
0.261
(0.876)
−0.171
(−0.967)
0.432
(1.499)
44
Table V
Economic Risk Premia: q ? Approach
I report coefficients of the economic risk premia on the following eight economic factors: IN F LU S , IN F LU K , IN F LJAP , IN F LGER , SP OT$/GBP ,
SP OT$/Y EN , SP OT$/DM , W LDM K. T-statistics, in parentheses, are adjusted for heteroskedasticity and serial correlation. The t-statistics refer
to the case where the composition of q ? is estimated outside and inside of the GMM algorithm, respectively. This table presents estimates of the
conditional premia when the set of returns is augmented to include managed portfolios. In this case, note that the intercept can be interpreted as the
unconditional risk premium with the larger set of securities.
Conditional Risk Premia
Risk Premia
IN F LU S
IN F LU K
IN F LJAP
IN F LGER
SP OT$/GBP
SP OT$/Y EN
SP OT$/DM
W LDM K
CON ST
−0.0128
−0.0198
−0.0145
0.0102
−0.0317
−0.0233
−0.0275
0.1171
(−1.12)
(−0.75)
(3.52)
(2.28)
JAN
−0.0680
0.0329
0.0258
0.0276
−0.0035
−0.0016
−0.0222
0.0895
0.0077
0.0069
0.0432
0.0565
0.0203
0.0381
0.0457
−0.1249
0.0313
−0.0948
−0.0666
−0.3286
−0.1126
0.2888
(−1.51)
(−1.02)
(2.65)
(1.67)
DIN F LU S
(−0.61)
(−0.45)
(−1.84)
(−1.54)
(0.26)
(0.17)
(−0.94)
(−0.85)
(0.81)
(1.11)
(0.30)
(0.20)
(−0.73)
(−0.62)
(1.16)
(0.93)
(1.49)
(1.06)
(0.45)
(0.48)
(0.64)
(1.15)
(1.59)
(1.43)
(−1.52)
(−0.95)
(−0.17)
(−0.12)
(0.89)
(0.74)
(−1.10)
(−0.63)
(−0.11)
(−0.05)
(1.41)
(1.22)
(−0.87)
(−0.57)
(1.28)
(1.35)
(1.59)
(1.56)
(−2.37)
(−2.20)
EU RO
−0.0149
0.0231
(−0.23)
(−0.18)
(0.34)
(0.35)
U SDIV Y LD
−0.0570
0.1027
0.0433
−0.0973
−0.1472
−0.3706
−0.1827
0.4617
(−2.16)
(−1.56)
(3.80)
(2.73)
W LDM K
−0.0280
0.0189
−0.0542
−0.0062
−0.0416
−0.0449
−0.0420
0.0331
(−1.31)
(−1.20)
(0.87)
(0.60)
0.0108
−0.0254
−0.0508
−0.0344
−0.0201
−0.0589
−0.0708
0.0527
DEF P REM
(−0.89)
(−0.72)
(−1.07)
(−0.88)
(0.32)
(0.29)
(1.48)
(1.48)
(0.64)
(0.67)
(−0.70)
(−0.84)
(0.53)
(0.41)
(0.66)
(0.52)
(−1.58)
(−1.57)
(−1.60)
(−1.74)
(−1.25)
(−1.13)
(−1.11)
(−1.04)
(−0.35)
(−0.23)
(−0.84)
(−0.70)
45
(−0.98)
(−0.72)
(−1.92)
(−1.50)
(−1.42)
(−1.36)
(−0.78)
(−0.60)
(−3.45)
(−2.09)
(−3.80)
(−2.44)
(−1.78)
(−1.20)
(−1.88)
(−1.34)
(−1.66)
(−1.39)
(0.82)
(0.91)
Table VI
Economic Risk Premia: y ? Approach
I report coefficients of the economic risk premia on the following eight economic factors: IN F LU S , IN F LU K , IN F LJAP , IN F LGER , SP OT$/GBP ,
SP OT$/Y EN , SP OT$/DM , W LDM K. T-statistics, in parentheses, are adjusted for heteroskedasticity and serial correlation. The t-statistics refer
to the case where the composition of q ? is estimated outside and inside of the GMM algorithm, respectively. This table presents estimates of the
conditional premia when the set of returns is augmented to include managed portfolios. In this case, note that the intercept can be interpreted as the
unconditional risk premium with the larger set of securities.
Conditional Risk Premia
Risk Premia
IN F LU S
IN F LU K
IN F LJAP
IN F LGER
SP OT$/GBP
SP OT$/Y EN
SP OT$/DM
W LDM K
CON ST
−0.0128
−0.0198
−0.0145
0.0102
(0.85)
(0.48)
−0.0233
−0.0275
0.1171
(−0.98)
(−0.62)
−0.0317
(−0.81)
(−0.75)
(2.28)
(2.28)
JAN
−0.0198
0.0244
−0.0252
0.0300
0.0108
0.0055
−0.0013
0.0859
0.0361
−0.0358
0.0075
−0.0102
0.0493
0.0931
0.0830
−0.1249
0.0310
−0.2296
−0.2157
−0.1636
0.3384
(−1.58)
(−1.52)
(1.97)
(1.97)
0.0091
−0.2806
−0.2693
−0.2259
0.5189
(−2.22)
(−2.11)
(3.07)
(3.07)
−0.0501
−0.0914
−0.0444
0.0364
(−1.06)
(−1.06)
(0.64)
(0.63)
−0.0439
−0.0358
−0.0675
0.0600
DIN F LU S
(−0.57)
(−0.45)
(−0.59)
(−0.46)
(1.09)
(0.92)
(−1.05)
(−0.85)
(0.68)
(0.65)
(−1.32)
(−1.16)
(−1.40)
(−0.97)
(0.50)
(0.33)
(1.79)
(1.24)
(−0.74)
(−0.47)
EU RO
−0.0805
0.0828
(−1.15)
(−0.89)
(1.22)
(1.01)
U SDIV Y LD
−0.1200
0.1027
0.0491
0.0298
−0.0007
0.0097
−0.0231
−0.0184
W LDM K
DEF P REM
(−1.72)
(−1.35)
0.0089
(0.31)
(0.27)
−0.0674
(−1.83)
(−1.52)
(1.52)
(1.24)
(1.18)
(1.06)
−0.0076
(−0.27)
(−0.22)
0.0320
(0.62)
(0.42)
(0.90)
(0.61)
(−0.03)
(−0.02)
(−1.22)
(−0.80)
(0.80)
(0.47)
(0.23)
(0.15)
(0.68)
(0.52)
(−0.98)
(−0.68)
46
(−0.95)
(−0.95)
(0.34)
(0.31)
(1.51)
(1.33)
(−2.31)
(−2.21)
(−2.79)
(−2.72)
(−1.23)
(−1.28)
(−0.96)
(−0.93)
(−0.65)
(−0.63)
(0.20)
(0.17)
(2.41)
(2.20)
(−1.92)
(−1.92)
(−2.41)
(−2.43)
(−1.84)
(−1.92)
(−0.72)
(−0.73)
(−0.04)
(−0.03)
(2.55)
(2.30)
(−1.28)
(−1.26)
(1.51)
(1.49)
(−2.24)
(−2.23)
(1.00)
(1.00)
Table VII
Unconditional Risk Premia, Volatility of Hedging Portfolios and Sharpe Ratios
I report unconditional economic risk premia (λ0 ), volatilities of the mimicking portfolios’ excess cash flows (v0 ) and Sharpe ratios
(Syk? ) commanded by the following economic factors: IN F LU S , INF LU K , IN F LJAP , IN F LGER , SP OT$/GBP , SP OT$/Y EN ,
SP OT$/DM , W LDMK. T-statistics, in parentheses, are adjusted for heteroskedasticity and serial correlation. The t-statistics
refer to the case where the composition of y ? is estimated outside the GMM algorithm.
Sharpe Ratios
IN F LU S
IN F LU K
IN F LJAP
IN F LGER
SP OT$/GBP
SP OT$/Y EN
SP OT$/DM
W LDM K
λ0
−0.0128
(−0.56)
−0.0198
(−1.03)
−0.0145
(−0.97)
0.0102
(0.84)
−0.0317
(−0.93)
−0.0233
(−0.63)
−0.0275
(−0.79)
0.1170
(2.18)
47
v0
0.4279
(16.06)
0.3563
(9.61)
0.2782
(17.60)
0.2262
(16.17)
0.6336
(21.75)
0.6846
(20.55)
0.6447
(18.63)
0.9947
(18.93)
Syk?
−0.0301
(−0.56)
−0.0558
(−0.98)
−0.0522
(−0.97)
0.0450
(0.85)
−0.0501
(−0.93)
−0.0339
(−0.63)
−0.0427
(−0.79)
0.1177
(2.10)
Table VIII
Tests of IS-CAPM, I-CAPM and II-CAPM
I perform conditional tests of the International Static CAPM (IS-CAPM), the International CAPM in presence of deviations
from PPP (I-CAPM (PPP)), the International CAPM in presence of currency risk (I-CAPM (SPOT)) and the international
intertemporal CAPM (II-CAPM(PPP) and II-CAPM(SPOT)) by using region subset tests (χ2 ), Hansen-Jagannathan variance bounds (HJV), and Hansen-Jagannathan distance measures (HJD), respectively. The benchmark standard deviation of
the scaled unrestricted normalized minimum—variance kernel is 0.3865. The standard deviations of the restricted normalized
?
?
?
?
minimum—variance kernels with conditional information are reported in column two. With qIS
, qI−P
P P , qI−SP OT , qII−P P P , and
?
qII−SP OT I denote the normalized minimum—variance kernels with conditional information for the IS-CAPM, I-CAPM (PPP),
I-CAPM(SPOT), II-CAPM (PPP), and II-CAPM(SPOT), respectively.
Models
Statistics
IS—CAPM
I-CAPM(PPP)
I-CAPM (SPOT)
II-CAPM(PPP)
II-CAPM(SPOT)
With conditioning information
χ2(dof )
(p−value)
HJV
(t−stat.)
std(qr? ) r=IS,I−P P P,I−SP OT,II−P P P,II−SP OT
33.09(27)
(0.19)
31.16(23)
(0.12)
31.17(24)
(0.14)
30.37(19)
(0.05)
30.42(20)
(0.06)
−0.2630
(−10.68)
? )=0.1231
std(qIS
−0.2038
(−7.48)
?
std(qI−P
P P )=0.1825
−0.2595
(−10.46)
?
std(qI−SP
OT )=0.1266
−0.1826
(−7.26)
?
std(qII−P
P P )=0.2037
−0.2549
(−10.21)
?
std(qII−SP
OT )=0.1313
48
HJD
(t−stat.)
0.3659
(12.62)
0.3402
(12.47)
0.3646
(12.43)
0.3280
(12.28)
0.3630
(12.35)
Table IX
Differences in Hansen-Jagannathan Variance Bounds and Distance Measures
I test whether the pricing implications delivered by the International Static CAPM (IS-CAPM), the International CAPM in
presence of deviations from PPP (I-CAPM (PPP)), the International CAPM in presence of currency risk (I-CAPM (SPOT))
and the international intertemporal CAPM (II-CAPM(PPP) and II-CAPM(SPOT)) are statistically different from each other.
The estimates of the differences in HJV and HJD measures are obtained by exactly identified GMM. T-statistics are obtained
using the delta method.
Statistics
IS—CAPM — I-CAPM(PPP)
IS—CAPM — I-CAPM(SPOT)
IS—CAPM — II-CAPM(PPP)
IS—CAPM — II-CAPM(SPOT)
I-CAPM(PPP) — I-CAPM (SPOT)
I-CAPM(PPP) — II-CAPM(PPP)
I-CAPM(PPP) — II-CAPM(SPOT)
I-CAPM(SPOT) — II-CAPM(PPP)
I-CAPM(SPOT) — II-CAPM(SPOT)
II-CAPM(PPP) — II-CAPM(SPOT)
(HJVr −HJVs )
(t−stat.)
std(qr? ) r,s=IS,I−P P P,I−SP OT,II−P P P,II−SP OT,r6=s
0.0592
(6.92)
0.0035
(1.85)
0.0804
(9.23)
0.0081
(2.89)
0.0557
(6.32)
0.0212
(2.89)
−0.0511
(−6.03)
0.0770
(8.52)
0.0047
(2.20)
0.0723
(8.13)
49
(HJDr −HJDs )
(t−stat.)
−0.0256
(−2.26)
−0.0012
(−0.62)
−0.0378
(−2.33)
−0.0028
(−0.79)
−0.0244
(−2.09)
−0.0122
(−1.57)
0.0227
(2.00)
−0.0366
(−2.19)
−0.0016
(−0.59)
−0.0350
(−2.11)
Table X
Hedging Demands
I report the unconditional normalized hedging demands for global market, inflation, and foreign exchange risks. T-statistics, in
parentheses, are adjusted for heteroskedasticity and serial correlation.
Hedging Demands
SP OT$/GBP
SP OT$/Y EN
SP OT$/DM
W LDMK
INF LU S
INF LU K
INF LJAP
INF LGER
50
α
0.129
(5.26)
0.038
(1.76)
0.058
(2.38)
0.109
(10.04)
0.222
(8.91)
0.364
(12.32)
0.173
(4.86)
−0.093
(−3.04)